Manifolds of low dimension with trivial canonical bundle in Grassmannians Vladimiro Benedetti To cite this version: Vladimiro Benedetti. Manifolds of low dimension with trivial canonical bundle in Grassmannians. Mathematische Zeitschrift, Springer, 2018. hal-01362172v3 HAL Id: hal-01362172 https://hal.archives-ouvertes.fr/hal-01362172v3 Submitted on 26 May 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Manifolds of low dimension with trivial canonical bundle in Grassmannians Vladimiro Benedetti∗ May 27, 2017 Abstract We study fourfolds with trivial canonical bundle which are zero loci of sections of homogeneous, completely reducible bundles over ordinary and classical complex Grassmannians. We prove that the only hyper-K¨ahler fourfolds among them are the example of Beauville and Donagi, and the example of Debarre and Voisin. In doing so, we give a complete classifi- cation of those varieties. We include also the analogous classification for surfaces and threefolds. Contents 1 Introduction 1 2 Preliminaries 3 3 FourfoldsinordinaryGrassmannians 5 4 Fourfolds in classical Grassmannians 13 5 Thecasesofdimensionstwoandthree 23 A Euler characteristic 25 B Tables 30 1 Introduction In Complex Geometry there are interesting connections between special vari- eties and homogeneous spaces. A striking evidence of this relation is the work of Mukai about Fano manifolds (as a reference, see [Muk88]). Mukai was able to reinterpret families of Fano threefolds as families of subvarieties of homogeneous spaces. His idea was that on a sufficiently general Fano threefold X, one can prove the existence of a certain positive vector bundle. Therefore, one gets a morphism, or eventually an embedding, from X to a certain Grassmannian. It ∗Aix-Marseille Universit´e, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France. 1 turned out that all the families of prime Fano threefolds (for which a classifica- tion was already known) admitted a nice description in terms of homogeneous bundles over Grassmannians. This description of the families was also helpful to understand better their geometry in relation to the well known geometry of the Grassmannians. From then, other works have showed the richness of homogeneous spaces in providing examples of special varieties. For instance, it is generally a dif- ficult problem to provide explicit families of hyper-K¨ahler manifolds, and few are known; among them, two can be seen as varieties of zeroes of a general global section of a homogeneous vector bundle over a Grassmannian. Both are fourfolds; the first one is the family of varieties of lines in a cubic fourfold, due to Beauville and Donagi ([BD85]). The variety of lines in a cubic fourfold is actually a subvariety of the Grassmannian of (projective) lines on a projective space of dimension five (Gr(2, 6)). It is the zero locus of a section of the third symmetric power of the dual of the tautological bundle. We denote this family of varieties by X1. The second one, more recent, is due to Debarre and Voisin ([DV10]). They take the Grassmannian Gr(6, 10) of 6-dimensional planes in a vector space V of dimension ten, and consider a general skew symmetric 3-form over V . The variety of planes isotropic with respect to this form is the zero locus of a section (which correspond to the form) of the third anti-symmetric power of the dual of the tautological bundle. They prove that this is a family, which we denote by X2, of fourfolds which are hyper-K¨ahler. These two examples motivated the present work. We study fourfolds which arise as zero loci of general global sections of homogeneous, completely reducible bundles over ordinary and classical Grassmannians. We will see that the only hyper-K¨ahler varieties of this form are those already mentioned; indeed, the following theorem holds: Theorem 1.1. Suppose Y is a hyper-K¨ahler fourfold which is the zero locus of a general section of a homogeneous, completely reducible, globally generated vector bundle over an ordinary or classical (symplectic or orthogonal) Grassmannian. Then either Y is of type X1 or of type X2. This theorem will be a direct consequence of the classification theorems of the following sections. For ordinary Grassmannians, we followed the analogous study done in [K¨u95], where the author has classified and then studied the properties of Fano fourfolds with index one obtained in the same way. Already in that case the two constraints for the varieties to be four dimensional and Fano of index one were sufficient to have a classification of the bundles which could give rise to the required varieties. We will generalize the result by giving a classification of fourfolds with triv- ial canonical bundle, following substantially the same ideas and proofs. With the help of the MACAULAY2-package SCHUBERT2 ([GS]) we will determine which subvarieties are Calabi-Yau (CY) and which are irreducible holomorphic symplectic (IHS, which is the same as hyper-K¨ahler) among the examples we have found. Then we will extend the classification to subvarieties of dimension four of the other classical Grassmannians. It should be remarked that, even though the symplectic and orthogonal Grassmannians can already be seen as varieties of zeroes of sections of homogeneous bundles over the ordinary Grassmannian, a new classification needs to be done. In fact, there are homogeneous bundles 2 over the classical Grassmannians that are not restriction of bundles over the ordinary ones. For instance, the orthogonal of the tautological bundle is not irreducible, and one can quotient it by the tautological bundle. Moreover, the spin bundles in the orthogonal case do not extend to a bundle on the ordinary Grassmannian. Finally we will present the corresponding results for dimension two and three. Some interesting varieties in this case have already been studied in detail. In particular, Mukai proved the unirationality of some moduli spaces of K3 surfaces with a given genus by giving an explicit locally complete family of K3s in a Grassmannian; those varieties are again zero locus of sections of homogeneous bundles ([Muk88], [Muk06], and [Muk92]). We give the classification for surfaces and threefolds and, for the surfaces, we report also the computation of the degree (which gives the genus of the natural polarization of the surface) and the Euler characteristic. Surprisingly enough, there are many more cases, and they would be worth to be studied thoroughly. As we were finishing the writing of this article, an article by D. Inoue, A. Ito and M. Miura which concerns the same subject has been published on arXiv ([IIM16]). In this work the authors proved that, under the same hypothesis as ours, a finite classification is possible for subvarieties of the ordinary Grassman- nian with trivial canonical bundle of any fixed dimension. They also study in more detail the case of CY threefolds, giving an explicit classification similar to ours and studying the cases found. On the other hand, they do not deal with the cases of symplectic and orthogonal Grassmannians, which is interesting too (for example, see Mukai’s articles on K3 surfaces of genus seven and eighteen). Acknowledgements. This work has been carried out in the framework of the Labex Archim`ede (ANR-11-LABX-0033) and of the A*MIDEX project (ANR- 11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Govern- ment programme managed by the French National Research Agency. I would like to thank my advisor Laurent Manivel for useful and numerous sug- gestions. Moreover, I would like to thank Alexander Kuznetsov for suggesting to include the case of OGr(n − 1, 2n) appearing in section 4.4, although this variety has Picard rank two. 2 Preliminaries In this section we recall some facts about homogeneous bundles over homoge- neous varieties; for a more complete exposition see [Ott95]. Afterwards, we recall basic definitions about varieties with trivial canonical bundle, and the Beauville-Bogomolov decomposition. 2.1 Homogeneous bundles Let G be a reductive complex algebraic group (for instance, one of the classi- cal groups SL(n, C), Sp(2n, C), SO(2n +1, C) or SO(2n, C)). A variety X is G-homogeneous if it admits a transitive algebraic (left) action of G. All ho- mogeneous varieties can be seen as quotients G/P of G by a subgroup P . A homogeneous variety G/P is projective and rational if and only if P contains a 3 Borel subgroup B (it is the case of the Grassmannians); in this case the subgroup P is said to be parabolic. Parabolic subgroups can be classified combinatorially. A homogeneous vector bundle F over a homogeneous variety X is a vec- tor bundle which admits a G-action compatible with the one on the variety X = G/P . If a vector bundle F is homogeneous, then the fiber F[P ] over the point [P ] ∈ X is stabilized by the subgroup P , i.e. F[P ] is a representation of P ; the converse holds as well. More precisely, there is an equivalence of cat- egories between homogeneous vector bundles over G/P and representations of P . Therefore, in this context, one can define irreducible and indecomposable homogeneous bundles, in analogy with the definitions in representation theory. Note that P , contrary to G, is not reductive in general.